KiDS-Legacy Shear: Cosmic Lensing Insights

• KiDS-Legacy shear is a comprehensive cosmic shear program that analyzes weak gravitational lensing through tomographic statistics across 967 deg², achieving percent-level precision on key parameters like S8.
• It utilizes advanced shape measurement techniques with lensfit calibration and self-organizing map (SOM) methods for redshift distribution, ensuring robust control of systematic biases.
• The program integrates real-space correlation functions, power-spectrum reconstructions, and thorough covariance analyses to facilitate rigorous cross-probe tests of structure formation and modified gravity.

The KiDS-Legacy shear program constitutes the ensemble of cosmic shear catalogs, analyses, and calibrated summary statistics derived from the Kilo-Degree Survey (KiDS), culminating in the DR5 "KiDS-Legacy" release. Cosmic shear refers to the weak gravitational lensing-induced distortion of background galaxy shapes, which encodes integrated information about the evolving matter distribution, geometry, and cosmic structure growth. KiDS-Legacy shear delivers tomographic two-point correlation functions, power-spectrum reconstructions, and internally validated data vectors suitable for cosmological inference, offering percent-level precision on $S_8 \equiv \sigma_8\sqrt{\Omega_m/0.3}$ and providing stringent cross-probe tests of structure formation, baryonic feedback, intrinsic alignments, and modified gravity.

1. Survey Data, Imaging, and Shape Measurement

KiDS-Legacy shear catalogs are derived from a 1347 deg² imaging footprint spanning the optical (u,g,r,i; VLT/OmegaCAM) and near-infrared (Z,Y,J,H,Kₛ; VIKING/VISTA) bands (Wright et al., 25 Mar 2025, Stölzner et al., 25 Mar 2025). After masking for bright stars, image defects, and edge effects, the usable area for cosmic-shear shape measurement is 967 deg², with an effective source galaxy density of $n_{\rm eff} \approx 8.8~{\rm arcmin}^{-2}$ and ellipticity dispersion $\sigma_e \approx 0.28$ per component (Wright et al., 25 Mar 2025, Broxterman et al., 10 Sep 2025). Shape measurements are performed using the lensfit forward model-fitting algorithm, calibrated on end-to-end multi-band SKiLLS image simulations (Li et al., 2022), yielding tomographic multiplicative biases $|m_{\rm final}| \lesssim 0.02$ and negligible additive residuals (Wright et al., 25 Mar 2025, Yoon et al., 1 Oct 2025). A robust "gold-weight" source selection prioritizes signal-to-noise, multiband photometric depth, and isolation (Zhang et al., 16 Jan 2026).

2. Redshift Distribution Calibration and Tomography

KiDS-Legacy utilizes six tomographic bins, defined on the Bayesian peak photo-z $z_B$ and spanning $0.10<z_B\leq2.0$, each with calibrated true redshift distribution $n^{(i)}(z)$ (Wright et al., 25 Mar 2025, Wright et al., 25 Mar 2025). Calibration is achieved via self-organizing map (SOM) colour-methods, leveraging a 126,085-object KiDZ spectroscopic compilation. To mitigate spectroscopic selection function and colour-space coverage biases, "prior-volume" weights and cross-correlation (CC) calibration against wide-area spectroscopic samples (DESI, SDSS, GAMA, VIPERS, 2dFLenS) are employed (Wright et al., 25 Mar 2025). Mean redshift biases per bin, $\delta z_i$, are $\lesssim 0.01$, and uncertainties have been reduced from KiDS-1000 values ($\sim0.02$) to $\sim0.01$ per bin (Wright et al., 25 Mar 2025), demonstrably tightening $S_8$ constraints (Wright et al., 25 Mar 2025, Stölzner et al., 25 Mar 2025).

3. Shear Correlation Functions, Power Spectra, and Covariance

The fundamental observables are the tomographic shear two-point correlation functions $\xi_+^{ij}(\theta)$ and $\xi_-^{ij}(\theta)$,

$$\xi_{\pm}^{ij}(\theta) = \frac{1}{2\pi} \int_0^\infty d\ell\, \ell\, J_{0/4}(\ell\theta)\, C_{\varepsilon\varepsilon}^{(ij)}(\ell),$$

where $C_{\varepsilon\varepsilon}^{(ij)}(\ell)$ is the sum of lensing (GG), intrinsic alignment (II, GI, IG) angular power spectra (Stölzner et al., 25 Mar 2025, Stölzner et al., 11 Dec 2025). Tomographic lensing kernels $W_i(\chi)$ incorporate the full source $n^{(i)}(z)$, with Limber-projected, nonlinear matter power spectrum $P_m(k,z)$ computed using HMCode2020, marginalizing over the baryon feedback parameter $\log_{10} T_{\rm AGN} \in [7.3,8.3]$ (Broxterman et al., 10 Sep 2025, Stölzner et al., 11 Dec 2025). Three summary statistics—real-space $\xi_\pm$, band-powers, and COSEBIs (Complete Orthogonal Sets of E/B-Integrals)—are used; COSEBIs provide lossless, finite-interval E/B separation for systematics control (Stölzner et al., 25 Mar 2025, Asgari et al., 2020, Asgari et al., 2018).

Covariance matrices are calculated analytically using the OneCovariance framework, which includes Gaussian cosmic variance, shape noise, non-Gaussian connected trispectrum (halo model), and super-sample covariance. Survey geometry and masking effects are incorporated to percent-level accuracy using measured data triplet/pair counts. Covariances among all summary statistics ($\xi_\pm, E_n, \mathcal B_n$) are modeled, enabling rigorous consistency and cross-probe tests (Reischke et al., 2024, Stölzner et al., 25 Mar 2025).

4. Calibration of Shear Systematics and Intrinsic Alignment Modeling

SKiLLS multi-band image simulations offer joint calibration of lensfit shear biases and redshift distributions. Residual multiplicative biases $m$ are determined per tomographic bin by matching observed galaxy S/N and morphology; blending-only and PSF-modeling systematics are sub-percent (Li et al., 2022, Yoon et al., 1 Oct 2025). PSF leakage $\alpha$, additive biases $c$, and position-dependent biases are controlled at the $10^{-3}$–$10^{-4}$ level, with grid-based diagnostics confirming spatial uniformity (Uitert et al., 2016).

Intrinsic alignments (IA) are modeled using an NLA-$M$ scheme: red galaxies (early-type/central) are assigned Gaussian priors $A_{\rm IA}^{\rm R}=3.2\pm0.5$, blue galaxies $A_{\rm IA}^{\rm B}=0.2\pm0.4$ (Johnston et al., 2018, Stölzner et al., 25 Mar 2025). Mass-dependence, redshift evolution, and luminosity scaling are incorporated if required by data splits (Stölzner et al., 25 Mar 2025). IA contributions (II/GI) are marginalized alongside baryon feedback, photometric-redshift shifts, and shear-calibration uncertainties in the cosmological likelihood.

5. Internal Consistency, Null Tests, and Cosmological Inference

KiDS-Legacy implements three tiers of internal consistency metrics—Bayes factor "suspiciousness," parameter-shift, and translated-posterior data-space tests—across splits in tomographic bins, scale, spatial region, and galaxy colour (Stölzner et al., 25 Mar 2025, Asgari et al., 2020). All splits except low-$\theta$ 2PCFs pass at $<1.4\sigma$; scale-split tension ($3.7\sigma$) is attributed to unmodeled baryonic feedback and non-Gaussian likelihoods (Stölzner et al., 25 Mar 2025). COSEBIs B-mode decompositions show no significant non-lensing B-modes, confirming the accuracy of PSF, photo-z, and additive bias corrections (Asgari et al., 2018). Position-dependent additive biases are $<3\times10^{-4}$ per chip and $<10^{-6}$ at tile scale—well below cosmic-shear signal (Uitert et al., 2016).

Power-spectrum deprojection reconstructs $P_m(k,z)$ over $k = 0.01$–$20~h~{\rm Mpc}^{-1}$, confirming $\Lambda$CDM at large scales ($\sigma_8=0.81$), with $30\%\pm10\%$ suppression at $k \gtrsim 3 h\,{\rm Mpc}^{-1}$ matching strong AGN feedback simulations (Broxterman et al., 10 Sep 2025).

6. Principal Cosmological Constraints and Legacy Impact

Joint KiDS-Legacy cosmic-shear analysis yields $S_8 = 0.815^{+0.016}_{-0.021}$ (KiDS alone), fully consistent with Planck CMB ($0.73\sigma$); combining KiDS-Legacy with DES Y3, Pantheon+ SNe Ia, and DESI Y1 BAO tightens $S_8$ to $0.814^{+0.011}_{-0.012}$ (Wright et al., 25 Mar 2025, Stölzner et al., 25 Mar 2025). The increased survey area, refined $n(z)$ calibration, and improved image reduction contribute a $\sim32\%$ gain in constraining power relative to KiDS-1000. Marginal degeneracies in $(\Omega_m, \sigma_8)$ are minimized by IA-informed priors and precise redshift calibration.

Analysis of minimal and extended cosmologies (curvature, neutrino mass, dynamic dark energy) demonstrates robust $S_8$ constraints: $S_8=0.813\pm0.018$–$0.816\pm0.006$ across all major probe combinations, with no residual tension with Planck (Reischke et al., 11 Dec 2025). Stringent bounds are placed on modified gravity parameters (e.g., Horndeski): $D_{\rm kin}=3.74^{+0.69}_{-1.92}$, $\Delta\hat{M}_*^2=0.32^{+0.07}_{-0.21}$, and effective Newtonian coupling deviation $\Delta\mu_{\infty,{\rm eff}}=0.066\pm0.023$ (Stölzner et al., 11 Dec 2025).

7. Cross-Correlations, Astrophysical Applications, and Future Prospects

KiDS-Legacy shear maps have enabled direct cross-correlation analyses with Fermi-LAT gamma-ray backgrounds, setting complementary WIMP annihilation and decay rate constraints in the low-GeV/TeV mass regime. Robust E/B decomposition, hybrid covariance estimation, and rigorous error propagation support weak-lensing applications well beyond dark energy and gravity tests (Zhang et al., 16 Jan 2026).

Systematic mitigation in shape measurement, blending, PSF modeling, and redshift distribution estimation are now fully integrated via SKiLLS simulation-based calibration, yielding percent-level control over systematic error budget and enabling reliable extension to Stage IV surveys. The release of summary statistics, covariance matrices, and posterior samples supports the use of KiDS-Legacy shear as a reference data set for future tomographic, large-scale structure, and multi-probe cosmological analyses (Reischke et al., 2024, Broxterman et al., 10 Sep 2025).